Deformations of associative Rota-Baxter operators

Abstract

Rota-Baxter operators and more generally $\\mathcal{O}$-operators on associative algebras are important in probability, combinatorics, associative Yang-Baxter equation and splitting of algebras. Using a method of Uchino, we construct an explicit graded Lie algebra whose Maurer-Cartan elements are given by $\\mathcal{O}$-operators. This allows us to construct a cohomology for an $\\mathcal{O}$-operator. This cohomology can also be seen as the Hochschild cohomology of a certain algebra with coefficients in a suitable representation. Next, we study linear and formal deformations of an $\\mathcal{O}$-operator which are governed by the above-defined cohomology. We introduce Nijenhuis elements associated with an $\\mathcal{O}$-operator which give rise to trivial deformations. As an application, we conclude deformations of weight zero Rota-Baxter operators and associative {\\bf r}-matrices.